The second article seems to imply that the teenager who overdosed (and died) on paracetamol/acetominophen (in 2011) was taking part in a "paracetamol challenge," but it's unclear (and doubtful, I think) that this was the case. (Yes, people do die from paracetamol/acetominophen overdose, however.) Of course, whether this is truly a thing is unclear as well.

It is funny (not funny ha-ha, funny strange) that while I am for the legalization (and hence regulation) of most recreational drugs, I believe that acetaminophen/paracetamol should be made Rx only. The active dose and the harmful/lethal dose are just way too close together. And too many people think that the way to get painkillers to work stronger or faster is to up the dosage. In reality, that can not only be very harmful, but counterproductive as well. A rebound headache is not an experience you want to go through. I had one once because of an inexperienced doctor telling me to take a couple Tylenol instead of a couple Advil* on top of my Rx painkillers for breakthrough pain. One of the worst headaches of my life, and I probably should have gone to the ER, but I couldn't bear the thought of a bumpy car ride.

*I do believe he unintentionally swapped the two name brands, probably due to being a half asleep/brain dead intern, and then didn't catch his own mistake.

I will only rarely take acetaminophen for pain or headaches precisely because of the dangers of overdosing. The idea that someone is challenging kids through social media to do this is appalling.
Question for UK readers: Is the Independent considered a reliable source? As a Boomer, I can remember alleged dangers from smoking banana peels etc. which may have been widely circulated ULs of their day.

I believe that acetaminophen/paracetamol should be made Rx only. The active dose and the harmful/lethal dose are just way too close together.

Ten acetaminophen pills taken at once will cause significant liver damage in most people. Up the dose a bit and it is lethal in some people. Up around 50 pills it starts to be lethal in nearly everyone taking that dose.

Making it Rx might help, especially since there are a number of other over the counter drugs with similar utility but without the toxicity. Or at least the labeling should be changed so that it is much clearer that a particular medication contains Acetaminophen. Vicodin as well as a fair number of other prescription and over the counter drugs contain it, see: common-medications

Question for UK readers: Is the Independent considered a reliable source?

It is, but they seem more or less just to be reporting a press release or warning from the police, plus a few quotes from a Mirror article. The Guardian is also reporting the warning from the police, but they've added that there's not much evidence that it's actually happening:

But while warnings have been issued, resulting in some dramatic media attention, there is little evidence that the dare has spread beyond a few isolated cases.

The hashtag #paracetamolchallenge is largely dominated by others urging young people not to take part in the fad. Some have shared personal experiences. One user wrote: “My sister died from taking too much paracetamol – kids don’t do it.”

Yes, the Independent's article was phrased rather carefully, noting (among other things), "a 'new social media craze'" (within quotes) and "reportedly encouraging" children and "allegedly spread" by social media, so it's fairly clear that editors there are reporting on warnings and panic associated with the belief that children are daring one another to take massive doses of the painkiller and maybe not so much about whether children are indeed doing so.

The Guardian's piece (thanks, RichardW!) also helpfully included mention of similar recent concerns about a "sleeper 'craze'" and "Neknominate" (which I hadn't heard of before), but the newspaper forgot to mention the "Game of 72," which had parents and school officials in a similar frenzy a month or so ago.